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A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

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Abstract

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

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References

  1. 1

    Alt H.W., Luckhaus S., Visintin A.: On nonstationary flow through porous media. Annali di Matematica Pura ed Applicata 136(1), 303–316 (1984)

  2. 2

    Arimburgo, F., Baiocchi, C., Marini, L.D.: Numerical approximation of the 1-D nonlinear drift-diffusion model in semiconductors. In: Nonlinear kinetic theory and mathematical aspects of hyperbolic system (Rapallo, 1992). Ser. Adv. Math. Appl. Sci., vol. 9, pp. 1–10. World Sci. Publ., River Edge, NJ (1992)

  3. 3

    Brezzi F., Marini L.D., Pietra P.: Méthodes d’éléments finis mixtes et schéma de Scharfetter–Gummel. C. R. Acad. Sci. Paris Sér. I Math. 305(13), 599–604 (1987)

  4. 4

    Brezzi F., Marini L.D., Pietra P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26(6), 1342–1355 (1989)

  5. 5

    Brézis H.: Analyse fonctionnelle: théorie et applications. Masson, Paris (1983)

  6. 6

    Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)

  7. 7

    Carrillo J.A., Toscani G.: Asymptotic L 1-decay of solutions of the porous medium equation to self-similarity. Indiana Univ. Math. J. 49(1), 113–142 (2000)

  8. 8

    Chainais-Hillairet C., Filbet F.: Asymptotic behavior of a finite volume scheme for the transient drift-diffusion model. IMA J. Numer. Anal. 27(4), 689–716 (2007)

  9. 9

    Chainais-Hillairet C., Liu J.G., Peng Y.J.: Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. M2AN 37(2), 319–338 (2003)

  10. 10

    Chainais-Hillairet C., Peng Y.J.: Convergence of a finite volume scheme for the drift-diffusion equations in 1-D. IMA J. Numer. Anal. 23, 81–108 (2003)

  11. 11

    Chainais-Hillairet C., Peng Y.J.: Finite volume approximation for degenerate drift-diffusion system in several space dimnesions. M3AS 14(3), 461–481 (2004)

  12. 12

    Courant R., Isaacson E., Rees M.: On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure. Appl. Math. 5, 243–255 (1952)

  13. 13

    Eymard R., Fuhrmann J., Gärtner K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006)

  14. 14

    Eymard R., Gallouët T.: H-convergence and numerical schemes for elliptic problems. SIAM J. Numer. Anal. 41(2), 539–562 (2003)

  15. 15

    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII. Handb. Numer. Anal., vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)

  16. 16

    Eymard R., Gallouët T., Herbin R., Michel A.: Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92, 41–82 (2002)

  17. 17

    Eymard R., Hilhorst D., Vohralík M.: A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105(1), 73–131 (2006)

  18. 18

    Il’in A.M.: A difference scheme for a differential equation with a small parameter multiplying the highest derivative. Math. Zametki 6, 237–248 (1969)

  19. 19

    Jüngel A.: Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. ZAMM 75(10), 783–799 (1995)

  20. 20

    Jüngel A.: Qualitative behavior of solutions of a degenerate nonlinear drift-diffusion model for semiconductors. Math. Models Methods Appl. Sci. 5(4), 497–518 (1995)

  21. 21

    Jüngel A., Pietra P.: A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 7(7), 935–955 (1997)

  22. 22

    Lazarov R.D., Mishev I.D., Vassilevski P.S.: Finite volume methods for convection–diffusion problems. SIAM J. Numer. Anal. 33(1), 31–55 (1996)

  23. 23

    Markowich P.A., Ringhofer C.A., Schmeiser C.: Semiconductor equations. Springer-Verlag, Vienna (1990)

  24. 24

    Markowich P.A.: The stationary semiconductor device equations, Springer edn. Computational Microelectronics, Vienna (1986)

  25. 25

    Markowich P.A., Unterreiter A.: Vacuum solutions of the stationary drift-diffusion model. Ann. Scuola Norm. Sup. Pisa 20, 371–386 (1993)

  26. 26

    Scharfetter D.L., Gummel H.K.: Large signal analysis of a silicon Read diode. IEEE Trans. Elec. Dev. 16, 64–77 (1969)

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Correspondence to Marianne Bessemoulin-Chatard.

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Bessemoulin-Chatard, M. A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme. Numer. Math. 121, 637–670 (2012). https://doi.org/10.1007/s00211-012-0448-x

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Mathematics Subject Classification (2000)

  • 65M12
  • 82D37